I worked the integral slightly differently, but your error isn't in the integral.

You have
$$I = \left. \frac{\pi}{4}\left[ \left(r^2 + 1\right) \ln{\left( r^2 + 1 \right)} - \left( r^2 + 1 \right) \right] \right|_0^3$$
Which I agree with. But the next line (the last one is wrong). I'm reasonably sure that you evaluated the expression at $r = 3$ correctly, but I think you got 0 at $r = 0$, which is not correct.